Partial regularity of stable solutions to the fractional Gel'fand-Liouville equation

TL;DR

This paper investigates the regularity of stable solutions to a fractional differential equation involving the Laplacian and exponential nonlinearity, establishing bounds on the dimension of their singular set.

Contribution

It provides a partial regularity result for stable solutions to the fractional Gel'fand-Liouville equation, extending understanding of singularities in fractional PDEs.

Findings

The dimension of the singular set is at most n-10s.

Stable solutions exhibit partial regularity depending on the fractional parameter s.

The results generalize classical regularity results to fractional operators.

Abstract

We analyze stable weak solutions to the fractional Gel'fand problem \begin{equation*} (-\Delta)^su=e^u\quad\mathrm{in}\quad \Omega\subset\mathbb{R}^n. \end{equation*} We prove that the dimension of the singular set is at most $n-10s.$